Boris Johnson and Jeremy Hunt have made their pitch to be the next prime minister at the first of 16 Conservative Party hustings.
The two contenders for Number 10 laid out their vision for the country at a conference in Birmingham.
Mr Johnson said these were "dark days" for his party, but insisted he could turn things around.
But his rival warned members not to elect the "wrong person" and risk "catastrophe".
Mr Johnson said the most important thing was to "get Brexit done".
He said: "My ambition is to unite this country and our society... let's take Britain forward.
"We need to discover a new confidence in our country."
'Slamming and banging'
The Metropolitan Police Service have said they will not be taking any further action following the episode.
Asked by the hustings moderator, LBC presenter Iain Dale, whether character mattered when choosing a prime minister, Mr Johnson said: "I don't think people want to hear about that."
Accused of ducking questions, Mr Johnson said: "People are entitled to ask me what I want to do for the country."
His rival, Mr Hunt, said the UK was in a "very serious situation".
He continued: "Get things wrong and and there will be no Conservative government, and maybe even no Conservative Party.
"Get things right and we can deliver Brexit, unite the party and send [Labour leader Jeremy] Corbyn packing."
But he warned that if Tory party members elected the "wrong person" as leader then "catastrophe awaits".
Mr Johnson, meanwhile, said he would prepare for a no-deal Brexit if he became PM.
He said: "We must be able to come out on WTO terms, so that for the first time in these negotiations we carry conviction.
"And it is precisely because we will be preparing between now and 31 October for a no-deal Brexit that we will get the deal we need."
He repeated his previous claim that it was "eminently feasible" for the UK to leave the EU by 31 October, saying he intended to make it happen.
That is the date that the EU's membership extension runs out, and if nothing has changed, the UK leaves without a deal.
Theresa May officially stood down as Tory leader on 7 June and will cease to be prime minister in the week commencing 22 July.
An initial list of 10 candidates to replace her was whittled down to Mr Hunt and Mr Johnson in a vote by Tory MPs.
In the fifth and final round on Thursday, Boris Johnson came out on top with 160 out of the 313 votes cast. Mr Hunt received 77 votes and Michael Gove was knocked out with 75.
One questioner at the hustings wanted to know whether Mr Johnson's approach to British business in the context of Brexit was as "cavalier and careless" as previously, when he used an expletive.
He replied: "I believe passionately in UK businesses, and as foreign secretary I spent a lot of my time promoting UK businesses at home abroad."
Jeremy Hunt insisted he would leave the EU with no deal if necessary.
He said: "I would do so with a heavy heart. But if we have to in the end I would do that."
Of a mooted renegotiation with Brussels, he said: "If we send the wrong person there's going to be no negotiation, no trust, no deal, and if Parliament stops that, maybe no Brexit.
"Send the right person and there's a deal to be done."
And challenged over the fact he campaigned for Remain in 2016, the would-be premier said: "Look at my record since that referendum.
"I have been very clear on every occasion... I have voted for Brexit."
'Our own Jeremy'
In another jibe at his rival, Mr Hunt warned members not to elect a Conservative "populist" to oppose "hard-left populist" Jeremy Corbyn.
Referring to himself, he said: "Or we could do better and choose our own Jeremy."
He continued: "If Corbyn gets into Downing Street there will never be Brexit.
"That's why it's so important that we hold together our Conservative and DUP family and deliver Brexit."
Mr Hunt said he would increase defence spending and called for Conservatives to have a "social mission", focusing on social care for older people.
He also vowed to get more young people voting Tory.
And he promised: "I will never provoke a general election before we have left the EU."
Members will receive their ballots between 6 and 8 July, with the new leader expected to be announced in the week beginning 22 July.